Stochastic Control Approach to Reputation Games

TL;DR

This paper applies stochastic control theory to reputation games, demonstrating how equilibrium payoffs can be characterized and computed under nested information structures, with implications for long-term strategic reputation management.

Contribution

It introduces a stochastic control framework for reputation games with nested information, establishing equilibrium characterization and payoff bounds under both discounted and undiscounted setups.

Findings

Equilibrium payoffs coincide with Markov Perfect Equilibrium payoffs.

Derived optimal strategies and payoff bounds for long-lived players.

Provided continuity results of equilibrium payoffs in prior beliefs.

Abstract

Through a stochastic control theoretic approach, we analyze reputation games where a strategic long-lived player acts in a sequential repeated game against a collection of short-lived players. The key assumption in our model is that the information of the short-lived players is nested in that of the long-lived player. This nested information structure is obtained through an appropriate monitoring structure. Under this monitoring structure, we show that, given mild assumptions, the set of Perfect Bayesian Equilibrium payoffs coincide with Markov Perfect Equilibrium payoffs, and hence a dynamic programming formulation can be obtained for the computation of equilibrium strategies of the strategic long-lived player in the discounted setup. We also consider the undiscounted average-payoff setup where we obtain an optimal equilibrium strategy of the strategic long-lived player under further technical conditions. We then use this optimal strategy in the undiscounted setup as a tool to obtain a tight upper payoff bound for the arbitrarily patient long-lived player in the discounted setup. Finally, by using measure concentration techniques, we obtain a refined lower payoff bound on the value of reputation in the discounted setup. We also study the continuity of equilibrium payoffs in the prior beliefs.